Special Lagrangian 3-folds are of interest in mirror symmetry, and in particular play an important role in the SYZ conjecture. One wishes to understand the singularities that can develop in families of these 3-folds; the relevant local model is provided by special Lagrangian cones in complex 3-space. When the link of the cone is a torus, there is a natural invariant g associated to the cone, namely the genus of its spectral curve. We show that for each g there are countably many real (g-2)-dimensional families of such special Lagrangian cones.